Applications of Non-Commutative Geometry

One of the central themes of modern mathematics has been the fusion of geometry and algebra. This began in the seventeenth century with Rene Descartes's remarkable insight that associated to any equation a geometric object; namely, the graph of the equation. Geometric properties of the graph encode algebraic properties of the equation, and vice versa. Throughout the eighteenth, nineteenth, and twentieth centuries, many mathematicians worked to deepen and extend Descartes's ideas. Thus the subject of algebraic geometry was developed. This subject has flourished and has achieved many outstanding results. In the mid-twentieth century--largely inspired by quantum theory and other physics--a new kind of algebra (operator algebras) emerged. More recently, this new algebra has been combined with geometry to form the new subject of noncommutative geometry. The mathematics of this project takes methods and results from noncommutative geometry and applies them to problems in the older, more traditional branches of mathematics.

The noncommutative geometry point of view has led to some startling conjectures and results. In the representation theory of reductive p-adic groups a totally unexpected geometric structure has been revealed. This greatly simplifies the representation theory and links the Baum-Connes conjecture (which is a conjecture within noncommutative geometry) to the Langlands program. For the index of geometrically-arising Fredholm operators, the noncommutative geometry point of view leads to the surprising conclusion that formulas like the Atiyah-Singer index formula apply well beyond elliptic operators. Hence ellipticity is not the essential point needed to obtain a topological formula for the index of such operators. This project will explore the many interactions of this wide-ranging set of ideas.